# nLab equivariant stable cohomotopy

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

The equivariant generalized cohomology theory which is represented by the equivariant sphere spectrum may also be called equivariant stable cohomotopy, as it is the equivariant stable homotopy theory version of stable cohomotopy, hence of cohomotopy. This is to be thought of as the first order Goodwillie approximation of plain (“unstable”) equivariant cohomotopy.

Just as the plain sphere spectrum is a distinguished object of plain stable homotopy theory, so the equivariant sphere spectrum is distinguished in equivariant stable homotopy theory and hence so is equivariant stable cohomotopy theory.

## Examples

### Of the point: The Burnside ring

###### Proposition

(Burnside ring is equivariant stable cohomotopy of the point)

Let $G$ be a finite group, then its Burnside ring $A(G)$ is isomorphic to the equivariant stable cohomotopy cohomology ring $\mathbb{S}_G(\ast)$ of the point in degree 0.

$A(G) \overset{\simeq}{\longrightarrow} \mathbb{S}_G(\ast) \,.$

This is due to Segal 71, a detailed proof is given by tom Dieck 79, theorem 8.5.1. See also Lück 05, theorem 1.13, tom Dieck-Petrie 78.

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point $\ast$
cohomology
of classifying space $B G$
(equivariant)
ordinary cohomology
HZBorel equivariance
$H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant)
complex K-theory
KUrepresentation ring
$KU_G(\ast) \simeq R_{\mathbb{C}}(G)$
Atiyah-Segal completion theorem
$R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$
(equivariant)
complex cobordism cohomology
MU$MU_G(\ast)$completion theorem for complex cobordism cohomology
$MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$
(equivariant)
algebraic K-theory
$K \mathbb{F}_p$representation ring
$(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$
Rector completion theorem
$R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{Rector 73}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant)
stable cohomotopy
K $\mathbb{F}_1 \overset{\text{Segal 74}}{\simeq}$ SBurnside ring
$\mathbb{S}_G(\ast) \simeq A(G)$
Segal-Carlsson completion theorem
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

More explicitly, this means that the Burnside ring of a group $G$ is isomorphic to the colimit

$A(G) \simeq \underset{\longrightarrow}{\lim}_V [S^V,S^V]_G$

over $G$-representations in a complete G-universe, of $G$-homotopy classes of $G$-equivariant based continuous functions from the representation sphere $S^V$ to itself (Greenlees-May 95, p. 8).

## References

Relation to Burnside ring

Relation to Segal-Carlsson completion theorem:

• Czes Kosniowski, Equivariant cohomology and Stable Cohomotopy, Math. Ann. 210, 83-104 (1974) (doi:10.1007/BF01360033 pdf)

• Erkki Laitinen, On the Burnside ring and stable cohomotopy of a finite group, Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 (jstor:24491306, pdf)

• Gunnar Carlsson, Equivariant Stable Homotopy and Segal’s Burnside Ring Conjecture, Annals of Mathematics Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 189-224 (jstor:2006940, pdf)

• Wolfgang Lück, The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups, Pure Appl. Math. Q. 1 (2005), no. 3, Special Issue: In memory of Armand Borel. Part 2, 479–541 (arXiv:math/0504051)

• Noe Barcenas, Nonlinearity, Proper Actions and Equivariant Stable Cohomotopy (arXiv:1302.1712)

A lift of Seiberg-Witten invariants to classes in circle group-equivariant stable cohomotopy is discussed in

• Stefan Bauer, Mikio Furuta A stable cohomotopy refinement of Seiberg-Witten invariants: I (arXiv:math/0204340)

• Stefan Bauer, A stable cohomotopy refinement of Seiberg-Witten invariants: II (arXiv:math/0204267)

• Christian Okonek, Andrei Teleman, Cohomotopy Invariants and the Universal Cohomotopy Invariant Jump Formula, J. Math. Sci. Univ. Tokyo 15 (2008), 325-409 (pdf)

Discussion of M-brane physics in terms of rational equivariant cohomotopy is in

Last revised on September 16, 2019 at 04:17:20. See the history of this page for a list of all contributions to it.